First order coherence function in terms of momentum distribution function Can someone show me how the first order coherence function $G^1(r,r')\equiv \left \langle \hat{\Psi}(r)\hat{\Psi}(r') \right \rangle $ for a system of bosons is related to the momentum distribution function $n(p)$, the diagonal elements of the density matrix in momentum representation, via: 
$G^1(r,r')=\frac{1}{V}\int dp n(p)exp\left [ \frac{i}{\hbar}p(r-r') \right ]$
My problem with the derivation given here (eq.2.27) is that I don't know where the delta function $\delta(p-p')$ assumed to reduce the integral over $p'$ comes from.
 A: I believe, that the derivation is wrong...
If you assume a translationally invariant state, such that $G^1(r, r') = G^1(r - r')$ then you can get the result. Rewrite the exponential as $p r - p' r' = p( r- r') + r'(p - p')$. Since, in this case, the left-hand-side of Eq. (2.27) can only depend on $r - r'$ it must be such that $p = p'$ from the second term. This gives you the delta-function in $(p - p')$ and you get the stated result.
To be more precise, the delta-function comes from $\langle \psi(p)^\dagger \psi(p')\rangle$ which can only depend on the relative momentum and is therefore proportional to $\delta(p - p')$ - by translational invariance.
I do not believe the result holds for states without translational invariance (such as a small, trapped ultra-cold quantum gas).
See for example Buus & Flensberg (http://www.amazon.com/Many-Body-Quantum-Theory-Condensed-Physics/dp/0198566336/ref=sr_1_2?ie=UTF8&qid=1366478600&sr=8-2&keywords=many+body+physics) appendix A.5 (the pages happens to be available in the amazon preview). 
